References: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 8.16.
Earlier, we analyzed the Stark effect in hydrogen using perturbation theory. The Stark effect causes a splitting of the spectral lines of hydrogen when an external electric field is applied. In our earlier post, we did a ‘proper’ analysis by using the correct Coulomb potential for the interaction between the proton and electron, but we can use a cruder model in which we treat the Coulomb attraction as a deep, but finite, square well with the bottom at zero and the top at an energy of . If the depth of the well satisfies (where is the width of the well) then the bound state energy levels are given by
The ground state energy is thus
Now suppose we add a weak electric field , so the electron feels a force (in the direction). To move the electron at constant speed from to we must apply a force (to prevent the electron from accelerating due to the electric field) over the interval so the work we do is
The potential energy due to is, if we take it to be zero at
Adding in this potential as a perturbation, the net potential for the electron is
The shape of this potential is a square well with a bottom that slopes downwards from left to right with a slope of , and a top that slopes downwards from a height of from onwards. Thus for any bound state energy , there is a point where and the particle could, in principle, tunnel out of the well and escape in the direction. (An escape in the direction isn’t possible since increases from as goes to the left of .)
and the barrier extends from to . In this case the barrier extends from to so we have
where the last line assumes .
In our study of alpha decay we got an estimate of the half-life of a particle as (equation 34 there)
where is the distance the electron must travel to reach the tipping point and is the speed of the electron. To get the speed of an electron in this potential we can take particle’s kinetic energy to be equal to its total energy, so
Plugging in the values given by Griffiths in the question:
As the age of the universe is around , this isn’t a tunneling event we can expect to see any time soon.